Podcast
Questions and Answers
When is the expression $\frac{51x^2 + 24x}{17x}$ undefined?
When is the expression $\frac{51x^2 + 24x}{17x}$ undefined?
What values of x make the expression $\frac{24x^2 + 9x}{8x + 3}$ undefined?
What values of x make the expression $\frac{24x^2 + 9x}{8x + 3}$ undefined?
What is the simplified form of $\frac{105}{42}$?
What is the simplified form of $\frac{105}{42}$?
List the two values of x that make the expression $\frac{x^3 - 9x}{x^2 + 5x + 6}$ undefined.
List the two values of x that make the expression $\frac{x^3 - 9x}{x^2 + 5x + 6}$ undefined.
Signup and view all the answers
What is the result of multiplying $\frac{12}{7}$ and $\frac{49}{36}$?
What is the result of multiplying $\frac{12}{7}$ and $\frac{49}{36}$?
Signup and view all the answers
A rational expression is defined when its denominator is zero.
A rational expression is defined when its denominator is zero.
Signup and view all the answers
Match the following rational expressions with when they are undefined:
Match the following rational expressions with when they are undefined:
Signup and view all the answers
A basic rule of fractions states that for any rational expression $\frac{ac}{bc}$, if _____ and _____, then $\frac{ac}{bc} = \frac{a}{b}$.
A basic rule of fractions states that for any rational expression $\frac{ac}{bc}$, if _____ and _____, then $\frac{ac}{bc} = \frac{a}{b}$.
Signup and view all the answers
Study Notes
Undefined Rational Expressions
- Rational numbers can be expressed as fractions of two integers; rational expressions are fractions of two polynomials.
- Denominators in rational expressions cannot equal zero to avoid undefined values.
- The domain of a rational expression includes all variable values except those that make the denominator zero.
- To find undefined values, set the denominator to zero and solve for the variable.
- Examples of expressions that are undefined:
- For ( \frac{51x^2 + 24x}{17x} ), undefined at ( x = 0 ).
- For ( \frac{24x^2 + 9x}{8x + 3} ), undefined at ( x = -\frac{3}{8} ).
- For ( \frac{x^3 - 9x}{x^2 + 5x + 6} ), undefined at ( x = -3, -2 ).
- For ( \frac{x^2 + 6x + 8}{x^2 - 16} ), undefined at ( x = -4, 4 ).
- The expression ( \frac{x^3 + 2x^2 + x + 2}{x^2 + 1} ) is defined for all real numbers as ( x^2 + 1 ) is never zero.
Simplifying Rational Expressions
- Simplification follows the rule: if ( \frac{ac}{bc} ) (where ( b ≠ 0 ) and ( c ≠ 0 )), then ( \frac{ac}{bc} = \frac{a}{b} ).
- To simplify, factor both the numerator and denominator, then divide out common factors.
- Example simplifications:
- ( \frac{105}{42} = \frac{5}{2} ) after factoring.
- ( \frac{14}{21} = \frac{2}{3} ).
- ( \frac{21}{39} = \frac{7}{13} ).
- ( \frac{15x}{13x} = \frac{15}{13} ).
- ( \frac{2x + 6}{3x + 9} = \frac{2}{3} ).
- ( \frac{x^2 + 9x + 14}{x^2 - 4} = \frac{x + 7}{x - 2} ).
- ( \frac{x^2 + 2x - 3}{x^2 - 5x + 4} = \frac{x + 3}{x - 4} ).
Multiplying Rational Expressions
- To multiply rational expressions, first factor the numerators and denominators completely.
- Identify and reduce common factors before multiplying.
- The Quotient Rule states ( \frac{x^a}{x^b} = x^{a-b} ) for any real numbers ( a ) and ( b ), under the condition ( x ≠ 0 ).
- Example of multiplication:
- ( \left(\frac{12}{7}\right)\left(\frac{49}{36}\right) = \frac{7}{3} ) after factoring and reducing.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz focuses on understanding undefined rational expressions and the conditions under which they become undefined. You'll learn how to identify these expressions and simplify them effectively. Test your knowledge with various examples and practice solving for the variable that causes the denominator to equal zero.
